\documentclass{article}
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\newtheorem{proposition}{Proposition}
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\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
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\newcommand\preoh[1]{\lnot #1}
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\begin{document}
\title{t37\_sin\_cos (TMKoEEqg6NYfCj4jUW1E9UXd1sau5whYta9)}
\maketitle
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_seq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k19\_sin\_cos : {\iota}$ be given.
Let $ k4\_series\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k23\_sin\_cos : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k16\_sin\_cos : {\iota}$ be given.
Let $ k22\_sin\_cos : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ v2\_comseq\_2 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k3\_series\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k4\_complex1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k11\_comseq\_3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_sin\_cos : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k7\_complex1 : {\iota}$ be given.
Let $ k3\_complex1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}((v2\_comseq\_2~(k3\_series\_1~\\
(k22\_sin\_cos~X0))){\unicdcdcdhd}((k4\_series\_1~(k22\_sin\_cos~X0)=k4\_complex1~\\
(k11\_comseq\_3~(k3\_sin\_cos~(k3\_xcmplx\_0~X0~k7\_complex1)))){\unicdcdcdhd}(\\
(v2\_comseq\_2~(k3\_series\_1~(k23\_sin\_cos~X0))){\unicdcdcdhd}(k4\_series\_1~(\\
k23\_sin\_cos~X0)=k3\_complex1~(k11\_comseq\_3~(k3\_sin\_cos~(k3\_xcmplx\_0~\\
X0~k7\_complex1)))))))\end{array}
\label{hyp:ehddgdpfdhjgogpfdgpgdh}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{({{X0} \in {X1}})}{\unicdbdjdcd}{({{{m1\_subset\_1}~{X0}}~{X1}})}}}}\label{hyp:ehbdpfdhfhcgdhfgeh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(v1\_funct\_1~k19\_sin\_cos){\unicdcdcdhd}((v1\_funct\_2~k19\_sin\_cos~k1\_numbers~\\
k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~k19\_sin\_cos~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k1\_numbers~k1\_numbers))))\end{array}
\label{hyp:egehpflgbdjdpfdhjgogpfdgpgdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(v1\_funct\_1~k16\_sin\_cos){\unicdcdcdhd}((v1\_funct\_2~k16\_sin\_cos~k1\_numbers~\\
k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~k16\_sin\_cos~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k1\_numbers~k1\_numbers))))\end{array}
\label{hyp:egehpflgbdgdpfdhjgogpfdgpgdh}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xreal\_0}~{X0}})}{\unicdbdjded}{({{X0} \in {k1\_numbers}})}}}\label{hyp:egbdpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k1\_numbers~k1\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers))))){\unicdbdjdcd}\\
((X0=k19\_sin\_cos){\unicdbdjded}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~k1\_numbers){\unicdbdjdcd}\\
(k1\_seq\_1~X0~X1=k3\_complex1~(k11\_comseq\_3~(k3\_sin\_cos~(k3\_xcmplx\_0~\\
X1~k7\_complex1))))))\end{array}
\label{hyp:egbdidpfdhjgogpfdgpgdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k1\_numbers~k1\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers))))){\unicdbdjdcd}\\
((X0=k16\_sin\_cos){\unicdbdjded}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~k1\_numbers){\unicdbdjdcd}\\
(k1\_seq\_1~X0~X1=k4\_complex1~(k11\_comseq\_3~(k3\_sin\_cos~(k3\_xcmplx\_0~\\
X1~k7\_complex1))))))\end{array}
\label{hyp:egbdgdpfdhjgogpfdgpgdh}
\end{equation}
\begin{theorem}\label{thm:ehddhdpfdhjgogpfdgpgdh}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((k1\_seq\_1~k19\_sin\_cos~X0=k4\_series\_1~\\
(k23\_sin\_cos~X0)){\unicdcdcdhd}(k1\_seq\_1~k16\_sin\_cos~X0=k4\_series\_1~(k22\_sin\_cos~\\
X0)))\end{array}$$
\end{theorem}
\end{document}
